On a chemotaxis model with nonlinear diffusion modelling multiple sclerosis

TL;DR

This paper proves the existence of global weak solutions for a chemotaxis model with nonlinear diffusion, which is used to simulate multiple sclerosis, highlighting the importance of nonlinear diffusivity in disease modeling.

Contribution

It introduces a chemotaxis model with nonlinear degenerate diffusion for multiple sclerosis and establishes the existence of global solutions under certain conditions.

Findings

Existence of global bounded solutions for the model.

Nonlinear diffusivity $D(m)$ is more appropriate for disease modeling.

Adaptation of existing strategies to prove solution existence.

Abstract

We investigated existence of global weak solutions for a system of chemotaxis type with nonlinear degenerate diffusion, arising in modelling Multiple Sclerosis disease. The model consists of three equations describing the evolution of macrophages ($m$), cytokine ($c$) and apoptotic oligodendrocytes ($d$). The main novelty in our work is the presence of a nonlinear diffusivity $D(m)$, which results to be more appropriate from the modelling point of view. Under suitable assumptions and for sufficiently regular initial data, adapting the strategy in [30,44], we show the existence of global bounded solutions for the model analysed.